Assumptions: Assume for simplicity that there are two genders and that each person has exactly one of those genders. Assume that the formal compatibility of a two-person relationship is purely a function of (1) each participant's proclivity for the two genders, and (2) the relation between the genders of the partners (i.e. same or different).

Statement of the problem: If you know the kinsey scores of me and two of my partners, estimate the compatibility of my two partners with each other.

Outline of the solution:

1. First, we define a numerical measure of a person's proclivity for each gender — a kind of kinsey scale.
2. Next, we define the formal compatibility of a person with their two-person relationship. It is a function of both the kinsey score of the individual, and whether they have the same or different gender from their partner. The overall compatibility of a relationship is basically the sum of each individual's compatibility with the relationship.
3. If you know the gender relationship between me and each of my two partners, you can easily deduce the gender relationship between them. When gender relations are represented by ±1, we can derive an arithmetic formula for this calculation.
4. Because this problem does not tell us the gender relations between the participants—only their kinsey scores—we must define a rule for estimating gender relationships between two people in a relationship given their kinsey scores.
5. Having defined a formula for estimating gender relationships given kinsey scores and a formula for computing formal compatibility, the expected compatibility is simply an expected-value calculation.

Suppose we measure proclivity/orientation using a “kinsey scale” ranging from -1 (exclusively prefers partners of the opposite gender) to +1 (exclusively prefers partners of the same gender)¹. Intermediate kinsey scores represent intermediate degrees of proclivity toward each gender, with a kinsey score of 0 representing equal proclivity toward each gender ("bi").

In the binary gender model, the formal relationship between the genders of any two people can be assigned pure kinsey-like value of ±1, corresponding to whether they have the same gender or different genders. I call this the gender polarity of the pair of people; any two people have a gender polarity, but the polarity is particularly relevant when the two people are in a relationship.

Suppose a person has a kinsey score of \(k\) and is involved in a relationship of polarity \(r=\pm 1\). How should we measure that person's individual compatibility with their relationship?

One simple choice of compatibility measure is \(C(r,k) = r\cdot k\), which assigns -1 compatibility to mismatched relationships (such as a person of purely opposite-gendered proclivity in a same-gendered relationship), and 1 to matched relationships (such as a person of purely different-gendered proclivity in a different-gendered relationship.

One defect with this measure is that it treats bi proclivity (i.e. a kinsey score of zero) as being middlingly compatible with any relationship \(C(r, 0) = 0\)—whereas it might be more desirable to have bi proclivity be fully compatible with any relationship. To patch this defect, we will instead use the slightly modified function \(C(r,k) = \min(1, 1 + r \cdot k)\) which assigns compatibility scores between 0 (mismatched proclivity and relationship) and 1 (matched proclivity and relationship), with bi proclivity receiving the maximal score along with other matched relationships.².)

Given two people in a relationship, their overall compatibility can be determined by adding up their compatibility scores. To place compatibility values on a -1 to 1 scale, we'll subtract 1 from the total.

$$C(r, k_1, k_2) = C(r,k_1) + C(r,k_2) - 1$$

In the next section, we will see that in a simplified version of the problem where you must determine my partners' compatibility with each other given full information—you know not only everyone's kinsey scores but also whether each pair of people has the same or different genders—the compatibility measure \(C(r, k_1, k_2)\) provides a direct, deterministic measure of my two partners' compatibility.

If you know whether I have the same gender as each of my partners, you can easily figure out whether my partners have the same gender or different genders from each other.

Numerically, if you know the gender polarity between me and each of my two partners — \(r_{1,2}, r_{1,3}\) — but don't know the relationship polarity \(r_{2,3}\) between my two partners, you can easily compute it:

$$r_{2,3} = r_{1,2}\cdot r_{1,3}$$.

To estimate the compatibility between my partners, you must somehow estimate the gender relations between them using our three kinsey scores. How should you predict whether a relationship is between same-gendered or different-gendered individuals as a function of the kinsey scores of the participants?

Let \(r(a, b)\) denote the probability that two people with respective kinsey scores of \(a\) and \(b\) are in a same-gendered relationship³. Based on common-sense considerations, the function \(r(a,b)\) should ideally satisfy the following criteria:

1. \(r\) is as smooth as possible, e.g. differentiable almost everywhere.

   Small changes in proclivity should never cause large changes in probability.

2. \(r\) is symmetric: \(r(a,b) = r(b,a)\).

   The order of the participants doesn't matter.

3. Inverting both kinsey scores inverts the probability: \(r(a,b) + r(-a,-b) = 1\)

   Inverting both kinsey scores instead gives the probability that they will be in an opposite-gendered relationship.

4. \(r\) is increasing: \(r(a+x, b) \geq r(a,b)\) when \(x > 0\).

   You can't decrease the probability of a straight relationship by making one of the participants straighter.⁴

5. \(r\) is maximal whenever both arguments are: \(r(1,1) = 1\)

   If the two participants have purely same-gendered proclivity, they are certain to be in a same-gendered relationship.

6. \(r\) is a linear interpolation whenever any argument is zero: \(r(a,0) = \frac{1}{2}(1+a)\).

   When any participant is bi, the probability that the relationship is between people of the same gender is dictated by the other participant.

Out of many possible functions with this behavior, one nice function is:

$$r(a,b) \equiv \min\left(1, \max \left(0, \frac{1+a+b}{2} \right)\right)$$

which is essentially the sum of the proclivities \(a\) and \(b\), rescaled to interpolate between 0 and 1 when proclivities are in a middle range, and cut off at 0 and 1 when proclivities are in an extreme range.

Now it's straightforward to compute the expected compatibility between my two partners given only everyone's kinsey scores:

$$\begin{eqnarray*}EC(k_1,k_2,k_3) &=& P(r_{2,3}=1) \cdot C(+1, k_2, k_3) + P(r_{2,3} = -1) \cdot C(-1, k_2, k_3)\\ \end{eqnarray*}$$

We can estimate the probability that r_2,3 = 1 by using the "love triangle polarity" formula above to express r_2,3 in terms of r_1,2 and r_1,3, then use the "relationship polarity from kinsey scores" to get probabilities for various values of r_1,2 and r_1,3 in terms of the kinsey scores of the participants.

$$\begin{eqnarray*} EC(k_1,k_2,k_3) &=& P(r_{2,3}=1) \cdot C(+1, k_2, k_3) + P(r_{2,3} = -1) \cdot C(-1, k_2, k_3)\\ &=&\small\left[r(k_1,k_2)\cdot r(-k_1,-k_3) + r(-k_1,-k_2)\cdot r(k_1,k_3)\right]C(+1, k_2, k_3) + \left[r(k_1,k_2)\cdot r(k_1,k_3) + r(-k_1,-k_2)\cdot r(-k_1,-k_3)\right]C(-1, k_2, k_3) \end{eqnarray*}$$

In order to solve our problem, we needed a way to estimate gender relationships from known kinsey scores. Assuming binary genders, the inverse problem of estimating kinsey scores from known gender relations results in a probability distribution isomorphic to drawing black/white balls out of an urn and representing your knowledge of the proportion of black/white balls in the urn — the binomial distribution, or the bi distribution for short.